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Transverse tau polarization in inclusive $\bar B\to τ\barνX$ decays

We calculate, in the framework of multi Higgs doublet models, the $CP$ violating transverse tau polarization, $P_\perp$, in inclusive semileptonic $B$ meson decays. We find that $P_\perp$ diverges at ${\cal O}(1/m_b^2)$ of the heavy quark expansion. We discuss the physical reasons for this divergence and show how to regularize it. We find large $1/m_b^2$ corrections that can suppress the $CP$ asymmetry by as much as 30\%. The maximal allowed asymmetry is $a_{CP}\sim0.34$. We discuss how the allowed range is expected to change in the future.