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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 

The Borel transform has the following properties: 

<img src='content_image/312992.jpg'>

<img src='content_image/312993.jpg'>

where n > 0. In Ref. [21] it was shown that the Borel transform is related to the inverse Laplace transform via 

<img src='content_image/312999.jpg'>

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 

<img src='content_image/312998.jpg'>

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 

