the discontinuity of the correlation function across the branch cut is required. As discussed in Chapter 1, due to the analytic properties of the correlation function, the discontinuity is related to the imaginary part. Therefore, once the correlation function has been calculated, the imaginary part must be extracted. This can be achieved via analytic continuation. The complexity of the functions that arise in QCD sum rule calculations depends on the number of external scales involved in the calculation. In the chiral limit, typically only logarithms involving dimensionless ratios of the Euclidean external momentum Q 2 and renor- malization scale µ occur. In order to deal with these, we define the complex logarithm as follows: Using this, we can define where we have considered a point below the branch cut, as described in Chapter 1. From (2.75) it can be shown that 76 Im log$^{[}$ Q 2 µ 2 ] = − π , Im log 2 $^{[}$Q 2 µ 2 ] = − 2 π log$^{[}$ t µ 2 ] , (2.76) log$^{[}$ Q 2 µ 2 ]∣ ∣ ∣ Q $^{2}$= te − $^{iπ}$, t> 0 ≡ log$^{[}$ t µ 2 ] − iπ, (2.75) log ( w ) ≡ log | w | + i Arg ( w ) , Arg ( w ) ∈ [ − π , π ) . (2.74) Figure 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. Im Q 2 Re Q 2