S U (3)

, the coefficients on the two sides of the bottom threshold are

β_{0}^{(4)} = 25/3

and

β_{0}^{(5)} = 23/3

. Continuity of the one-loop running of the strong coupling at

m_{b}

gives

\frac{25}{3} ln \frac{m _{b}^{2}}{Λ _{4}^{2}} = \frac{23}{3} ln \frac{m _{b}^{2}}{Λ _{5}^{2}}, ln \frac{m _{b}}{Λ _{5}} = \frac{25}{23} ln \frac{m _{b}}{Λ _{4}} .

(1)

Thus matching the QCD scale across a quark threshold yields

Λ_{5} = m_{b} (\frac{Λ _{4}}{m _{b}})^{25/23} = Λ_{4} (\frac{m _{b}}{Λ _{4}})^{- 2/23}, p = 2, r = 23.

(2)

The 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.