This thesis reports the first search for $Bc+\to D(\ast )0\mu +\nu \mu$ decays using proton-proton collision data collected by the LHCb experiment operating at the Large Hadron Collider at CERN. The measurement makes use of a data sample collected between 2016 and 2018 at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of $5.1fb-1$. The semi-exclusive signal $Bc+\to D(\ast )0\mu +\nu \mu$ branching fraction is measured relative to the normalisation decay, $Bc+\to J/\psi \mu +\nu \mu$. The branching fraction ratio is found to be

$B(Bc+\to D(\ast )0\mu +\nu \mu )B(Bc+\to J/\psi \mu +\nu \mu )\mathrm{=0.0405\pm 0.0082\pm 0.0065\pm 0.0004,}$

and is exploited to extract the first-ever measurement of the CKM ratio, $|Vub|/|Vcb|$, using semileptonic $Bc+$ decays,

$|Vub||Vcb|=0.200\pm 0.020\pm 0.041\pm 0.001.$

In both results, the first uncertainty is statistical, the second is systematic, and the third is due to external measurements of the $D0\to K-\pi +$ and $J/\psi \to \mu +\mu -$ decay branching fractions. This proof-of-concept measurement demonstrates the potential for CKM metrology with the rare $Bc+$ meson.

In addition, this thesis details the development of a novel deep learning architecture to perform calorimetric shower reconstruction at high energy particle physics experiments. A bespoke network, exploiting recent developments in image recognition and geometric deep learning, is designed to achieve one-shot 2D cluster reconstruction capable of learning an arbitrary detector-plane geometry. The performance of the detection network is assessed using a standalone dataset inspired by the LHCb Electromagnetic Calorimeter layout.