as masses, in terms of QCD parameters. However, in practice this approach fails. In general we are interested in the ground state hadron in a certain J PC channel. The spectral function will include this state, along with excited states and the continuum. Hence it is difficult to isolate the ground state contribution when such a dispersion relation is used. In addition, the correlation function Π ( Q $^{2}$) often contains field theoretical divergences and its value at Q 2 = 0 is usually unknown. The critical insight of Refs. [106, 107] is that these difficulties can be overcome by applying the Borel transform to the dispersion relation (1.79), which is defined as ˆ B ≡ lim N, Q 2 →∞ N/Q $^{2}$≡ τ ( − Q $^{2}$) N Γ( N ) ( d d Q 2 ) N . (1.80) ˆ B $^{[}$Q 2 n ] = 0 , ˆ B$^{[}$ Q 2 n t + Q 2 ] = τ ( − 1) $^{n}$e − tτ , (1.81) The Borel transform has the following properties: where n > 0. In Ref. [21] it was shown that the Borel transform is related to the inverse Laplace transform via where b is defined such that f ( Q $^{2}$) is analytic to the right of the integration contour. Multi- plying both sides of Eq. (1.79) by ( − Q $^{2}$) k and taking the Borel transform using Eq. (1.81), the dispersion relation becomes Note that the Borel transform has removed the Π (0) and Π $^{′}$(0) terms. In addition, any terms in the explicit field-theoretic expression for Π ( Q $^{2}$) that are polynomials in Q 2 will be removed by the Borel transform. Note that this includes any divergences of the form ϵ − $^{n}$f ( Q $^{2}$) where f ( Q $^{2}$) is a polynomial in Q $^{2}$. However, such terms will not be eliminated ˆ B τ [ $^{(}$− Q 2 ) k Π $^{(}$Q 2 ) ] = 1 π ∫ ∞ t$_{0}$ dt t $^{k}$e − tτ ρ $^{had}$( t ) . (1.83) ˆ B τ $^{[}$f $^{(}$Q 2 )] = L − 1 $^{[}$f $^{(}$Q 2 ) ; τ ] = 1 2 πi b + i ∞ ∫ b − i ∞ dQ $^{2}$f $^{(}$Q 2 $^{)}$e Q $^{2}$τ , (1.82) 37