content_image/301519.jpg content_image/301527.jpg content_image/301528.jpg
## 5.5 Improvements 

<img src='content_image/301519.jpg'>

Since the publication of [2] in December 2005, there have been many further publications, suggesting improvements to the simple model presented in the preceding parts of this chapter. We consider here one of the more well- known extensions, achieved by including a dilaton in the action (5.2) and generalizing the metric (5.1) to: 

<img src='content_image/301527.jpg'>

<img src='content_image/301528.jpg'>

As yet, Φ( z ) and A ( z ) are undetermined. We can constrain them with the following pieces of physics: 

1. The geometry must return to AdS in the UV ( z → ϵ ), so the linear combination Φ − A must asymptote to log z for small z 

2. A simple flux model [117] shows that meson masses obey the relation m 2 n ∼ n for each radial excitation n . This agrees with experiment (figure 5.1). To reproduce this phenomenon, Φ − A must asymptote to z 2 for large z . It is worth noting that the model [2] we analysed in the previous section predicts a meson mass spectrum with m 2 n ∼ n 2 

