This thesis presents the results of a time-dependent analysis of $B^0\to D^{\mp}\pi^{\pm}$ decays using $3~\rm fb^{-1}$ of
proton-proton collision data collected
with the LHCb detector at CERN's Large Hadron Collider during Run 1 with a centre-of-mass energy of $7$ (2011) and $8$ (2012) TeV.
The LHCb experiment is dedicated to the study of the properties
of $b$-flavoured hadrons, in particular $CP$ violation in the $B$ meson system.
The Standard Model of Particle Physics
describes very precisely the mechanism and the amount of $CP$ violation expected in the Universe.
However, the observed matter-antimatter asymmetry is larger by several order of magnitude
compared to the predictions. This could be explained by the existence of a new source of $CP$ violation, originating in
New Physics beyond the Standard Model.
The time-dependent analysis of $B^0\to D^{\mp}\pi^{\pm}$ decays provides constraints on
the angle $\gamma$ of the Unitarity Triangle, one of the fundamental parameters
of the Standard Model related to $CP$ violation. Since no sizeable high-order Standard Model processes are expected to contribute,
any deviation from the predictions would be an unambiguous signature
of New Physics.
The current experimental precision on $\gamma$ is significantly lower than that of theoretical predictions.
This motivates the effort for new experimental determinations of $\gamma$ in order to reduce its uncertainty.
The analysis of $\Bz\to\Dmp\pipm$ decays, although not as sensitive as that obtained from decays of
charged $B$ mesons into $D^{()0}K^{()+}$ final states, represents an independent and uncorrelated estimation of $\gamma$
that contributes to the global combination of all $\gamma$ measurements. The result obtained in this thesis is more precise than previous
determinations from other experiments (BaBar, Belle) using $B^0\to D^{\mp}\pi^{\pm}$ decays.
Although based on a very large sample of about half a million signal events, it is still dominated by statistical uncertainties,
indicating good prospects for future improvements in precision with more data from Run 2 and beyond.
In addition to the $B^0\to D^{\mp}\pi^{\pm}$ analysis, this thesis also summarizes the studies to improve the
performances of the flavour tagging algorithms used by the LHCb collaboration to infer
the flavour of neutral $B$ mesons in time-dependent analyses.
The performance of these algorithms, being correlated with the kinematics of the reconstructed particles
as well as the complexity of the event (number of tracks and primary vertices), showed a significant
decrease on Run 2 data (2015--2018), which were collected at a centre-of-mass energy of $13~\rm TeV$.
Thanks to new implementations, these algorithms now have a performance similar to that
obtained with Run 1 data.