x 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 1.10 Examples of stable (left) and unstable (right) sum rules. The stable sum rule exhibits τ stability within the sum rule window whereas the unstable sum rule does not. In the unstable case there is no region in the sum rule window where the mass prediction given by Eq. (1.113) displays weak dependence on the Borel parameter. Mass predictions made from unstable sum rules are unreliable for this reason. The left and right figures correspond to the scalar and pseudoscalar charm-light diquark sum rules from Chapter 4, respectively. 48 2.1 Feynman diagram representing the A -type massive tadpole integral. . . . . . 54 2.2 Integration contours for the k$_{0}$ integral in (2.7). Poles are indicated by ⊗ symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3 Feynman diagram representing the B -type self-energy integral. . . . . . . . . 56 2.4 Feynman diagram representing the V -type two-loop nested integral. . . . . . 59 2.5 Feynman diagram representing the J -type two-loop sunset integral. . . . . . 62 2.6 Feynman diagram representing the F -type two-loop master integral. . . . . . 64 2.7 A branch cut singularity in the complex Euclidean momentum plane. The branch cut is on the interval Q 2 ∈ ( −∞ , − t$_{0}$ ], where t$_{0}$ is the hadronic threshold. 76 B.1 The Gamma function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107