Worked Examples & Physical Applications
The R-ratio, e⁺e⁻ machines, numerical evaluations, and experimental confrontation
4.19 The R-Ratio
The R-ratio is defined as the ratio of the hadronic cross section to the muon pair production cross section:
At energies well above quark thresholds but below the Z pole, quark pair production dominates the hadronic cross section. Each quark flavor $q$ with charge $Q_q$contributes:
where $N_c = 3$ is the number of colors. Summing over all accessible quark flavors:
Computing $R$ for different energy ranges (where $Q_u = 2/3$, $Q_d = Q_s = -1/3$, $Q_c = 2/3$, $Q_b = -1/3$):
| Energy Range | Active Quarks | R (leading order) |
|---|---|---|
| $\sqrt{s} < 2m_c \approx 3$ GeV | u, d, s | $3(4/9 + 1/9 + 1/9) = 2$ |
| $2m_c < \sqrt{s} < 2m_b$ | u, d, s, c | $3(4/9 + 1/9 + 1/9 + 4/9) = 10/3$ |
| $\sqrt{s} > 2m_b \approx 10$ GeV | u, d, s, c, b | $3(4/9 + 1/9 + 1/9 + 4/9 + 1/9) = 11/3$ |
💡R-Ratio as a Quark Counter
The R-ratio is one of the most direct experimental proofs of the quark model and the existence of color. Without color ($N_c = 1$), R would be 2/3 below charm threshold — the measured value of 2 requires $N_c = 3$. Each new quark threshold produces a step increase in R, beautifully confirming the sequential discovery of charm and bottom quarks.
4.20 Numerical Cross Section Evaluations
Worked Example 1: e⁺e⁻ → μ⁺μ⁻ at LEP
At LEP-II with $\sqrt{s} = 200$ GeV (far from the Z pole), the QED cross section is:
Converting units using $\hbar c = 0.197$ GeV·fm and $1\,\text{fb} = 10^{-39}\,\text{cm}^2$:
In practice, the conversion factor $(\hbar c)^2 = 0.3894$ mb·GeV$^2$ is essential. A useful numerical formula:
Worked Example 2: Thomson scattering numerical value
The Thomson cross section with $r_e = \alpha/m_e$:
Evaluating: $r_e = 2.818 \times 10^{-13}$ cm, so:
4.21 e⁺e⁻ Colliders and QED Tests
Electron-positron colliders have been the precision workhorses of QED and the Standard Model. The key machines and their QED-relevant measurements:
SPEAR/SLAC ($\sqrt{s} \sim 3{-}8$ GeV)
Discovered the $J/\psi$ (c&cmacr;) and τ lepton. Measured R-ratio confirming$N_c = 3$. The step from $R = 2$ to $R = 10/3$ at charm threshold was dramatic evidence for the charm quark.
PEP/PETRA ($\sqrt{s} \sim 10{-}46$ GeV)
Discovered the gluon via 3-jet events. Tested the $(1+\cos^2\theta)$ angular distribution of e⁺e⁻ → μ⁺μ⁻ to high precision, confirming spin-1/2 for quarks.
LEP/CERN ($\sqrt{s} = 91{-}209$ GeV)
Precision Z-pole measurements. Bhabha scattering used for luminosity (0.05% precision). At energies away from the Z, QED predictions for $e^+e^- \to \mu^+\mu^-$ were verified to sub-percent accuracy. Deviations from pure QED at $\sqrt{s} = m_Z$revealed the Z boson contribution.
💡QED as the Precision Benchmark
QED predictions serve as the "null hypothesis" against which new physics is tested. Any deviation from QED at e⁺e⁻ colliders signals either weak interaction effects (which modify cross sections near $\sqrt{s} \sim m_Z$) or genuinely new physics (contact interactions, extra dimensions, etc.). The agreement of QED with experiment to ~10 significant digits (in the electron g-2) makes it the most precisely tested theory in all of science.
4.22 Forward-Backward Asymmetry
The angular distribution $d\sigma/d\cos\theta \propto (1 + \cos^2\theta)$ for$e^+e^- \to \mu^+\mu^-$ is symmetric under $\theta \to \pi - \theta$. The forward-backward asymmetry is defined as:
In pure QED, $A_{\text{FB}} = 0$ because the $(1+\cos^2\theta)$ distribution is forward-backward symmetric. However, when weak interaction effects are included (Z boson exchange), the distribution acquires a $\cos\theta$ term:
At the Z pole, $A_{\text{FB}} \approx 0.017$ for muons. This small but nonzero value, arising from the parity-violating Z coupling, was one of the key measurements at LEP that precisely determined the weak mixing angle $\sin^2\theta_W$.
4.23 Leading QED Corrections
The tree-level results we derived receive radiative corrections. The leading $O(\alpha)$correction to $e^+e^- \to \mu^+\mu^-$ modifies the total cross section:
The correction $3\alpha/(4\pi) \approx 0.17\%$ comes from vertex corrections and vacuum polarization. In practice, a more important effect is initial state radiation(ISR), where the electron or positron radiates a photon before annihilating:
where $f(x,s)$ is the radiator function and $xs$ is the reduced CM energy squared after radiation. This effect reduces the effective $\sqrt{s}$ and must be carefully corrected in all precision measurements.
Key Concepts (Page 4)
- • R-ratio: R = N_c Σ Q_q² directly measures the number of colors and active quark flavors
- • Numerical: σ(e⁺e⁻ → μ⁺μ⁻) = 86.8 nb/(s/GeV²), Thomson cross section = 0.665 barn
- • e⁺e⁻ colliders (SPEAR, PEP, LEP) confirmed QED to extraordinary precision
- • Forward-backward asymmetry A_FB = 0 in pure QED; nonzero values signal weak interactions
- • Leading QED correction: 3α/(4π) ≈ 0.17% to the total cross section
- • Initial state radiation reduces effective √s and must be corrected experimentally