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## 6.2 Results 

The emphasis of this chapter is on the field-theoretic aspects of the publication above. As mentioned in Chapter 1, multiple currents may couple to a single hadronic state. For instance, consider a state | h 〉 that couples to both scalar meson and glueball currents: 

<img src='content_image/310652.jpg'>

Hadrons that couple to multiple currents can be studied within QSR using non-diagonal cor- relation functions. In this case the non-diagonal correlation function contains scalar glueball and quark meson currents 

<img src='content_image/310653.jpg'>

<img src='content_image/310654.jpg'>

This correlation function can be calculated using the perturbative expansion (1.19) and the OPE (1.99) as usual. However, the leading order contribution to the perturbative Wilson coefficient contains a non-local divergence. Because this divergence arises at leading order, it cannot be canceled through a multiplicative renormalization. 

This problem can be solved by considering the renormalization of the composite operator representing the scalar glueball current, which mixes with the scalar meson current under renormalization. The renormalized scalar glueball operator is given by 

<img src='content_image/310655.jpg'>

where n$_{f}$ is the number of quark flavours and the subscripts R and B denote renormalized and bare quantities, respectively. [97, 89]. The renormalized scalar glueball operator must be used in order to renormalize the non-diagonal correlation function (6.3). Note the appearance of the second term in Eq. (6.4) which is divergent and proportional to the scalar meson current. This arises due to operator mixing and must be included in the QSR analysis. This term amounts to a renormalization-induced contribution to the non-diagonal correlation 

