by the Borel transform when the function f ( Q $^{2}$) is not a polynomial in Q $^{2}$. These are called non-local divergences and must be dealt with through renormalization. Furthermore, the Borel transform has introduced an exponential factor which serves to suppress excited state contributions to the hadronic spectral function. In order to isolate the ground state contribution, it is conventional to parametrize the hadronic spectral function in terms of a resonance and continuum: where θ ( t − s$_{0}$ ) is the Heaviside step function and s$_{0}$ is the continuum threshold ( s$_{0}$ > t$_{0}$ ). The continuum contribution is related to the imaginary part of the QCD correlation function through the optical theorem [100]. Inserting this into Eq. (1.83) yields The quantity R$_{k}$ ( τ , s$_{0}$ ) can be calculated in QCD, and is related to the spectral function ρ $^{res}$( t ). The spectral function ρ $^{res}$( t ) can be measured experimentally, or it can be modeled in terms of the physical properties of the hadron being studied. Therefore, Eq. (1.86) provides a direct relationship between QCD calculations and hadronic parameters. This is the central identity of QCD Laplace sum rules. Before proceeding it is useful to consider possible forms that R$_{k}$ ( τ , s$_{0}$ ) can take. Typi- cally, the correlation function Π ( Q $^{2}$) involves functions that have a branch cut on the interval Q 2 ∈ ( −∞ , − t$_{0}$ ] and functions that have a pole at Q 2 = − t$_{0}$ . Those that have a pole generally have the form where n is a positive integer. Because Eq. (1.87) has no imaginary part, the contribution of 38 Π pole $^{(}$Q 2 ) ∼ 1 ( Q 2 + t$_{0}$ ) n , (1.87) R$_{k}$ ( τ , s$_{0}$ ) = 1 π ∫ ∞ t$_{0}$ dt t k e − tτ ρ $^{res}$( t ) , (1.85) R$_{k}$ ( τ , s$_{0}$ ) = ˆ B τ [ $^{(}$− Q 2 ) k Π $^{(}$Q 2 ) ] − 1 π ∫ ∞ s$_{0}$ dt t k e − tτ ImΠ $^{(}$te − iπ ) . (1.86) ρ $^{had}$( t ) = ρ $^{res}$( t ) + θ ( t − s$_{0}$ ) ρ $^{cont}$( t ) , ρ $^{cont}$( t ) = ImΠ $^{(}$te − iπ ) , (1.84)