= Matching the QCD scale across a quark threshold
At a heavy-quark mass $m$, leading-order continuity gives $\beta_- \ln(m/\Lambda_-)=\beta_+\ln(m/\Lambda_+)$. Therefore $\boxed{\Lambda_+=m(\Lambda_-/m)^{\beta_-/\beta_+}}$. This is a change of the <strong-coupling scale> parametrization across a change of active flavour number, not a discontinuity of the matched <QCD coupling>. For four to five active flavours in $SU(3)$, $\beta_-=25/3$ and $\beta_+=23/3$, hence $\Lambda_5=\Lambda_4(m_b/\Lambda_4)^{-2/23}$.
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