High power $n$ of $m_b$ in $b$-flavored widths and $n=5\to \mathbf{\infty }$ limit

The leading term in the semileptonic width of heavy flavor hadrons depends on the fifth power of the heavy quark mass. We present an analysis where this power can be self-consistently treated as a free parameter $n$ and the width can be studied in the limit $n\to \infty .$ The resulting expansion elucidates why the small velocity (SV) treatment is relevant for the inclusive semileptonic $b\to c$ transition. The extended SV limit (ESV limit) is introduced. The leading terms in the perturbative ${\alpha }_s$ expansion enhanced by powers of $n$ are automatically resummed by using the low-scale Euclidean mass. The large-$n$ treatment explains why the scales of order $m_b/n$ are appropriate. On the other hand, the scale cannot be too small since the factorially divergent perturbative corrections associated with running of ${\alpha }_s$ show up. Both requirements are met if we use the short-distance mass normalized at a scale around $m_b/n\sim 1\mathrm{GeV}.$ A convenient definition of such low-scale operator-product-expansion-compatible masses is briefly discussed.