$V_{cb}$ determination from inclusive $b \to c$ decays: an alternative method

The determination of $V_{cb}$ relies on the Heavy-Quark Expansion and the extraction of the non-perturbative matrix elements from inclusive $b\to c$ decays. The proliferation of these matrix elements complicates their extraction at $1/m_b^4$ and higher, thereby limiting the $V_{cb}$ extraction. Reparametrization invariance links different operators in the Heavy-Quark expansion thus reducing the number of independent operators at $1/m_b^4$ to eight for the total rate. We show that this reduction also holds for spectral moments as long as they are defined by reparametrization invariant weight-functions. This is valid in particular for the leptonic invariant mass spectrum ($q^2$), i.e. the differential rate and its moments. Currently, $V_{cb}$ is determined by fitting the energy and hadronic mass moments, which do not manifest this parameter reduction and depend on the full set of 13 matrix elements up to $1/m_b^4$. In light of this, we propose an experimental analysis of the $q^2$ moments to open the possibility of a model-independent $V_{cb}$ extraction from semileptonic decays including the $1/m_b^4$ terms in a fully data-driven way.