Figure 1.7: Integration contour used to derive the dispersion relation between the correlation function and its imaginary part. The correlation function has a branch cut on the interval z ∈ ( −∞ , − t$_{0}$ ], where t$_{0}$ is the hadronic threshold. which is depicted in Fig. 1.7. The branch cut singularity is due to the correlation function and the integrand has poles at z = 0 and z = Q 2 as depicted in the figure. The value of n is chosen to ensure that the contribution of the radial contour vanishes as its radius is taken to infinity. As an example, we will derive the dispersion relation used in Chapter 4, where the correlation function satisfies The dispersion relation can be derived by evaluating the contour integral (1.72) in two ways and equating the results. First, we will evaluate the contribution from each portion of the contour C . The radial portion of the contour can be bounded using (1.73): where we have set n = 2 to ensure that the contribution of the radial contour is zero. In order to determine the contribution of the portion of C that circles the branch point, we must know the behaviour of the correlation function near the hadronic threshold t$_{0}$ . The correlation function in Chapter 4 is regular at this point, therefore the contribution of the portion of the contour that circles the branch point is zero. The only remaining portions of the contour C are those that are above and below the branch cut in Fig. 1.7. For these, we lim z $_{→∞}$Π ( z ) ∼ z log $^{2}$( z ) . (1.73) | I$_{R}$ | ≤ lim R →∞ Π ( R ) R n = lim R →∞ R log $^{2}$( R ) R n = lim R →∞ log $^{2}$( R ) R = 0 , (1.74) 35