Meson mass spectrum How to calculate the meson spectrum in this background has been detailed in [62]. We summarise the results here. The meson spectrum corresponds to x-dependent fluctuations of the fields u$_{5}$ and u$_{6}$ on the D7-brane world volume. To find the spectrum, the mesons are treated as small fluctuations on the brane surface, meaning that the DBI action can be expanded to second order, with no interaction between the meson fields. Hence we can treat the fields as freely moving plane waves, which will allow us to explicitly calculate the spectrum. There are two directions in which the fields u$_{5}$ and u$_{6}$ can fluctuate: perpendicular to the ρ -dependent flow of the brane, and parallel to the ρ -dependent flow. We use the U(1) symmetry to write this as The induced metric on the D7 brane can be written as with ρ 2 = u 2 − L $^{2}$. L is the distance between the centres of the D7 brane and the D3 brane stack. It is therefore proportional to the quark mass: the constant of proportionality is unimportant in this analysis. It just means that we will predict the masses of the mesons relative to one another, rather than relative to some absolute scale. The DBI action is then given by 52 u$_{5}$ = m + χ ( x, r ) u$_{6}$ = φ ( x, r ) (3.10) ds 2 = ρ 2 + L 2 R 2 η$_{µν}$dx $^{µ}$dxν + R 2 ρ 2 + L $_{2}$dρ 2 + R $^{2}$ρ2 ρ 2 + L $_{2}$d Ω 2 3 (3.11) S$_{DBI}$ = − T$_{7}$ ∫ d $^{8}$ξ √ − det G ( 1 + 2( $_{πα}$′$_{R}$$_{)}$ 2 g cd ρ 2 + L $_{2}$( ∂$_{c}$χ∂$_{d}$χ + ∂$_{c}$φ∂$_{d}$φ ) ) (3.12)