Figure 3.3: Numerical calculation of the meson spectrum in AdS$_{5}$ × S $^{5}$. The dots are the analytical values. 2 4 6 8 M GLYPH<Minus> 1.0 GLYPH<Minus> 0.5 0.5 1.0 g GLYPH<LParen1> M GLYPH<RParen1> 3. Where the value of g ( M ) changes sign indicates a stable flow: an example plot for (3.13) with L = 1 , M = √$_{2 is shown in figure 3.3}$ Figure 3.3 shows that the numerics agree with the analytical solution. This will be important - in future we will not be able to find an analytical solution, so we have to be confident that the numerical solutions can be trusted. To give an even clearer idea of what is happening around one of the normalizable solutions to (3.13) we plot f ( r ) for three different values of M around √$_{2 in figure 3.4. It can be seen that as we pass through}$$_{M}$ = √$_{2}$ the solution flips from −∞ to + ∞ , so the only normalizable solution in this neighbourhood is at M = √$_{2.}$ We analysed the scalar meson sector in section 3.2.1. Now we re-introduce the U(1) gauge field that we neglected in equation (3.5): 3.2.2 Vector Meson Spectrum 55 S$_{D}$$_{7}$ = − T$_{7}$ ∫ d $^{8}$ξ √ − det ( P [ g ] + 2 πα ′ F ) (3.19)