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Can the differences in the determinations of $V_{ub}$ and $V_{cb}$ be explained by New Physics?

The precise determination of the CKM elements $V_{cb}$ and $V_{ub}$ is crucial for any new physics analysis in the flavour sector. Their values can be determined from several tree-level decays: $V_{cb}$ can be extracted from $B\to D\ellν$ and $B\to D^*\ellν$ while $V_{ub}$ can be obtained from $B\to π\ellν$, $B\to ρ\ellν$ and $B\toτν$. In addition, for both $V_{cb }$ and $V_{ub}$ an inclusive determination is available. There is a long lasting discrepancy between the inclusive and exclusive determinations which recently even increased for $V_{cb}$ above the $3\;σ$ level. In this article we study the possible effect of new physics on the inclusive and exclusive determination of $V_{cb}$ and $V_{ub}$ in a model independent way. We find that there is only one operator corresponding to a modified $W$ coupling which can achieve this. However, respecting $SU(2)$ gauge invariance at the high scale this would lead to very large violations of the $Z$ to $b\bar b$ coupling not compatible with experiment. Therefore, we conclude that a new physics explanation of the difference between the inclusive and exclusive determination of $V_{cb}$ and $V_{ub}$ is currently ruled out. Therefore, the discrepancies must be due underestimated uncertainties in the theoretical and/or the experimental analysis.