The large number of scalar meson states below 2 GeV contradicts the expected number derived from a quark-antiquark description. One possibility is that one or more of the light scalar mesons can be described as four quark states composed of quark-antiquark pairs. This scenario has been explored with sum-rule methods in Quantum Chromodynamics (QCD) to leading-order in the strong coupling constant. Higher loop contributions are significant in the QCD sum-rule analysis of quark-antiquark scalar states and a similar situation could occur in the four-quark case. In this thesis the leading order and pieces of the second order terms of the correlation function, as needed to study properties of a four-quark state via a QCD sum-rule, is calculated in the chiral limit (i.e. massless quarks) in QCD. Operator mixing related to renormalization of the composite operators appearing in the correlation function first contributes at second order. The result for the second order contributions to the correlation function indicate that operator mixing must be addressed before using proper dispersion relations to link this calculation with the mass of an existing state.

The large number of scalar meson states below 2 GeV contradicts the expected number derived from a quark-antiquark description. One possibility is that one or more of the light scalar mesons can be described as four quark states composed of quark-antiquark pairs. This scenario has been explored with sum-rule methods in Quantum Chromodynamics (QCD) to leading-order in the strong coupling constant. Higher loop contributions are significant in the QCD sum-rule analysis of quark-antiquark scalar states and a similar situation could occur in the four-quark case. In this thesis the leading order and pieces of the second order terms of the correlation function, as needed to study properties of a four-quark state via a QCD sum-rule, is calculated in the chiral limit (i.e. massless quarks) in QCD. Operator mixing related to renormalization of the composite operators appearing in the correlation function first contributes at second order. The result for the second order contributions to the correlation function indicate that operator mixing must be addressed before using proper dispersion relations to link this calculation with the mass of an existing state.