The ratio R = sigma(e+e- -> hadrons)/sigma(e+e- -> mu+mu-) at center-of-mass energies well above quark thresholds but below the Z pole is predicted by the quark model. For five quark flavors (u, d, s, c, b), what value does the quark model predict, and what does it tell us about color?
AR = 5 (one for each quark flavor)
BR = N_c * sum of e_q^2 = 3 * (4/9 + 1/9 + 1/9 + 4/9 + 1/9) = 3 * 11/9 = 11/3 -- the factor of 3 comes from color, and the charges-squared sum over the five active flavors
CR = 1 because all quarks produce the same final state (hadrons)
DR = 3 (one for each color)
Each quark flavor contributes e_q^2 to the cross section (from the photon-quark coupling), multiplied by N_c = 3 for color. The sum over u, d, s, c, b gives (4/9 + 1/9 + 1/9 + 4/9 + 1/9) = 11/9, times 3 = 11/3 approximately 3.67. QCD corrections modify this to R = 11/3 * (1 + alpha_s/pi + ...). The measurement of R = 3.67 +/- small corrections, rather than R = 11/9 = 1.22 (which would result without color), was direct evidence for three colors.
Question 2 Short Answer
In 1979, the TASSO experiment at PETRA observed three-jet events in e+e- annihilation. These events were the first direct evidence for the gluon. How does a third jet arise from the process e+e- -> hadrons?
Think about your answer, then reveal below.
Model answer: At leading order, e+e- -> qqbar produces a quark-antiquark pair that hadronizes into two back-to-back jets. At next-to-leading order in QCD, one of the quarks can radiate a hard gluon: e+e- -> qqbar-g. If the gluon has sufficient energy and is emitted at a large angle, it hadronizes into a third jet distinct from the quark and antiquark jets. The rate and angular distribution of three-jet events are predicted by QCD (proportional to alpha_s) and confirmed the gluon as a spin-1 particle. The angular correlations between the three jets distinguish a spin-1 gluon from hypothetical spin-0 gluons.
The discovery of three-jet events was a landmark in particle physics. It confirmed not only the existence of the gluon but also its vector (spin-1) nature, consistent with being the gauge boson of SU(3). The four-jet rate (e+e- -> qqbar-gg or qqbar-qqbar) provided additional tests of the non-abelian structure of QCD.
Question 3 Multiple Choice
At center-of-mass energy equal to the Z boson mass (91.2 GeV), the ratio R increases dramatically to approximately 20. Why?
ANew quark flavors become accessible at this energy
BThe Z boson resonance enhances the cross section enormously -- e+e- -> Z -> qqbar dominates over the photon diagram, and the Z couples to quarks with both vector and axial-vector couplings that are larger than the photon's electromagnetic coupling
CQCD corrections become very large at 91 GeV
DThe quarks become asymptotically free and the cross section diverges
At the Z pole, the process e+e- -> Z -> hadrons dominates. The Z couples to all quarks (including qqbar pairs with charge 0 total), and its coupling strength (proportional to the weak mixing angle and the quark quantum numbers) greatly exceeds the electromagnetic coupling. The Z pole cross section is sigma ~ 12*pi/M_Z^2 * Gamma_e*Gamma_had/Gamma_tot^2. The LEP experiments at CERN made precision measurements of the Z lineshape, extracting the number of light neutrino species (N_nu = 2.984 +/- 0.008) from the total width.