R-ratio

R is the ratio of the hadronic cross-section to the muon cross-section in electron-positron collisions.

$R=\frac{\sigma (e^{+}{e}^{-}\to hadrons)}{\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})}$.

At leading order in the strong coupling constant ${\alpha }_s=g_s^2/4\pi$, the $e^{+}{e}^{-}$ can only decay to a $q\bar{q}$ pair. Then,

$R=N_c{\sum }_q{Q}_q^2\theta (\sqrt{s}-2m_q)$,

where the index $q=(u,d,s,\dots )$ runs over the six flavours of quarks, $\theta$ is the Heaviside step function, $\sqrt{s}$ is the centre of mass energy of the $e^{+}{e}^{-}$ pair and $m_Q$ are the quark masses. The quark charges are $Q_u=Q_c=Q_t=2/3$ and $Q_d=Q_s=Q_b=-1/3$. The Heaviside function ensures that until the centre of mass energy reaches the threshold for the pair production of the given quark, it is not produced and hence doesn’t contribute to the R-ratio.

• The experimental measurement shows some resonant peaks, corresponding to the production of bound-state quarks or on-shell production of hadrons.
• The leading order (LO) prediction does reasonably describe the data away from these resonances, from which we can extract the number of quarks and colours.
• There is a slight excess in the data compared to the LO prediction, which can be accounted for by higher order/radiative corrections.