evaluation of leading order contributions to the Wilson coefficients involves the calculation of multiple two-loop momentum integrals. These loop integrals are often quite difficult to evaluate and constitute a significant technical barrier to extending QSR calculations to higher orders. Chapter 2 discusses techniques for evaluating loop integrals. As mentioned previously, the hadronic spectral function can be measured experimentally. For instance, the spectral function for hadronic states with J PC = 1 −− is related to the ratio of the cross sections This spectral function is shown in Fig. 1.9. Figure 1.9: The hadronic spectral function R ( s ). The horizontal axis is the center of mass frame collision energy of the electron and positron in units of GeV and the vertical axis is the dimensionless number R ( s ). The resonances labeled ρ , ω and φ correspond to distinct hadrons. The electron and positron annihilate through a virtual photon or Z boson, both of which have the quantum numbers J PC = 1 $^{−−}$. Therefore all of these hadrons must have these quantum numbers. The horizontal location of each resonance peak is the mass of the hadron corresponding to the resonance. The region between 1 . 5 GeV and 3 . 0 GeV is the continuum which is described well by the three-loop perturbative QCD calculation. Note that in the region below 1 . 5 GeV the QCD prediction and resonance features agree in the sense of a global average. This is an example of the concept of quark-hadron duality which is crucial to QSR. Figure taken from Ref. [20]. 46 1.4.4 Hadronic Spectral Function R ( s ) = σ ( e $^{−}$e + → hadrons ) σ ( e $^{−}$e + → µ $^{−}$µ $^{+}$) . (1.110)