Semi-leptonic decays at the Belle and Belle II experiments

This thesis explores semi-leptonic decays of the type $B \to V L$ along two avenues; their contribution to high-precision tests and metrology of the Standard Model, and their potential for probing physics beyond the Standard Model. The magnitude of the Cabbibo-Kobayashi-Maskawa matrix element corresponding to $b\to c$ quark transitions, $|V_{cb}|$, has a long-standing discrepancy between its measurements from exclusive and inclusive decays, which presents a puzzling gap in our knowledge of the Standard Model. In this thesis, recent Lattice QCD calculations of form factors at nonzero hadronic recoil are incorporated into novel analysis methods to examine the phenomenology and analysis techniques behind the exclusive $|V_{cb}|$ measurement in $B \to D^{*}\ell^+ \nu_\ell$ decays. This results in a measurement of $|V_{cb}|=(38.49 \pm 0.54 \pm 0.92 \pm 0.91) \times 10^{-3}$, the least model-dependent result to date. The results of this analysis of the $B \to D^{*}\ell^+ \nu_\ell$ branching fraction and decay rate are used in a test of Standard Model QCD predictions of hadronic factorisation, with the first measurement of the parameter $|a_1|$ within a single experiment, based on a ratio that cancels most experimental systematic uncertainties, resulting in $|a_1| = 0.884 \pm 0.004 \pm 0.003 \pm 0.016$. This corresponds to an $8.9\sigma$ discrepancy from the Standard Model, suggesting the existence of large non-factorisable contributions in hadronic $B$-decays, or potential contributions from new physics amplitudes. A probe for new physics in photon-dipole interactions, described by the $\mathcal{O}_7$ term in an operator product expansion, is then performed to constrain non-Standard Model right-handed currents in flavour-changing neutral current semi-leptonic decays. New lepton identification techniques in the Belle experiment are developed for use in the first study of $B \to K^* e^+ e^-$ decays at the very low dilepton invariant mass region, $q^2 < 1.12 \gevccsq$, to co