 Solution

For the Quantum chromodynamics gauge group $SU(3)$, the coefficients on the two sides of the bottom threshold are ${\beta }_0^{(4)}=25/3$ and ${\beta }_0^{(5)}=23/3$. Continuity of the one-loop running of the strong coupling at $m_b$ givesThus matching the QCD scale across a quark threshold yieldsThe scale parameter changes because the beta-function coefficient changes, even though the matched coupling is continuous. This is leading-order matching with a fixed flavour number within each region; the stated four-flavour region is the effective region above the charm threshold, not a literal claim that four quarks remain active down to arbitrarily small scales.