Past exam of the particle physics course of the University of Oxford / 2022 / 3 / c / 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 isWith $N_c=3$, 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 $e^{+}{e}^{-}\to {\gamma }^{\ast }\to q\bar{q}g$, 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.

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.