Observable Measured (MeV) Best Fit (MeV) m$_{π}$ 140 141 m$_{ρ}$ 776 832 m$_{a}$$_{1}$ 1230 1220 f$_{π}$ 92 84 F 1 / 2 ρ 345 353 F 1 / 2 a 1 433 440 Table 5.2: Results of model from [2] for QCD observables. m$_{q}$ = 2 . 3MeV, σ = (327MeV) $^{3}$, z$_{m}$ = 323(MeV) − 1 5.5 Improvements 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: As yet, Φ( z ) and A ( z ) are undetermined. We can constrain them with the following pieces of physics: 80 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 S = ∫ d 5 $_{x e}$− Φ( z ) √ g Tr { | DX | 2 + 3 | X | 2 − 1 4 g 2 5 ( F 2 L + F 2 $_{R}$) } (5.25) ds 2 = e 2 A ( z ) ( dz 2 + η$_{µν}$dx $^{µ}$dxν ) (5.26)