8 Chapter 1: Introduction Figure 1.3: R = σ ( e $^{+}$e − → hadrons ) /σ ( e $^{+}$e − → µ $^{+}$µ $^{−}$) plotted against electron energy in GeV. Above 5 GeV, R is well approximated by N$_{c}$ $^{∑}$Q 2 f with N$_{c}$ = 3 . Figure adapted from [7]. e $^{+}$e − depends on N$_{c}$ ; again, experimentally N$_{c}$ = 3. Similarly, the inclusive semi-hadronic decay rate of the τ lepton to its semi-leptonic one is a direct measure of N$_{c}$ with N$_{c}$ = 3. And the decay of neutral pions to two photons is a direct measure of the square of the number of colors; the theoretical prediction where f$_{π}$ = 92 . 4 MeV is the pion decay constant controlling the π − → µν , is in remarkable agreement with data (7 . 7 ± 0 . 6 eV). Four jet measurements identified directly the triple gluon vertex and found that the gauge group is SU(3) instead of SO(3) or perhaps U(1)$_{3}$ [43, 44]; see Fig. 1.4. Similarly the running of the coupling agrees well for N$_{c}$ = 3 [66]. Γ( π 0 → γγ ) = [ N$_{c}$ ( Q 2 u − Q 2 $_{d}$)] 2 ( α 2 64 π 2 ) m 3 π f 2 π = 7 . 7 eV , (1.1)