= Solution
Away from narrow <resonances> and <quark> thresholds, the <QED> annihilation <cross section> into each accessible <quark flavour> is its squared fractional <electric charge> times the <muon>-pair <cross section>, multiplied by the number $N_c$ of distinguishable <color charges>. Thus the inclusive hadronic ratio is
$$
R_{\rm had}=\frac{\sigma(e^+e^-\to\text{hadrons})}{\sigma(e^+e^-\to\mu^+\mu^-)}
\simeq N_c\sum_f Q_f^2.
$$
With $N_c=3$, the continuum predictions are $R_{uds}=2$, $R_{udsc}=10/3$ and $R_{udscb}=11/3$ as successive <charm quark> and <bottom quark> channels become accessible. The measured normalization and steps support three <color charges>. At leading order these hadronic events form two back-to-back <jets> from the <quark> and <antiquark>; quantitative comparisons include <QCD> corrections and avoid the nonperturbative threshold region.
The extra resolved <jet> is explained by $e^+e^-\to\gamma^*\to q\bar qg$, with <gluon> radiation from either the <quark> or <antiquark>. One possible <color charge> routing is initial $q_r\bar q_{\bar r}$ followed by $q_r\to q_b+g_{r\bar b}$; the final $q_b$, $\bar q_{\bar r}$ and $g_{r\bar b}$ combine to a net <colour singlet>.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2022-colour-processes.png]
{title=Colour flow in a three-jet process and in gluon-fusion Higgs production}
The three-<jet> <cross section> contains one additional <strong interaction> vertex in the amplitude, so its ratio to the two-<jet> rate is of order $\alpha_s=g_s^2/(4\pi)$ times a resolution-dependent kinematic coefficient. A resolved fraction of order $15\%$ is consistent with perturbative <QCD> and a <strong coupling constant> much larger than the <fine-structure constant>. It is not an equality $\alpha_s=0.15$: the <jet> definition and cuts are needed. <Photon> radiation has a much smaller electromagnetic rate. The large hadronic third-<jet> rate supports <gluon> radiation; identifying the radiated object specifically as a colour-octet <gluon> also uses the <QCD> vertex and colour-flow description, not merely the existence of three visible <jets>.
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