5.3 Calculations 5.3.1 Vector Mesons There are now only three parameters in our model: z$_{m}$, m$_{q}$ and σ . The hadrons of QCD correspond to the normalizable modes of the 5D fields. Requiring the modes to be normalizable means that when we substitute the chosen modes back into the 5D action, the action remains finite. As usual, this means that the solution must die away sufficiently rapidly in the UV, and be smooth over the IR boundary (here given by z = z$_{m}$ ). This gives us two explicit conditions for a general solution ψ ( z ). There is also a third constraint: that of orthogonality. ψ$_{n}$ and ψ$_{m}$ must be orthogonal to one another if they are to represent distinct mesons. So we also have ∫ ( dz/z ) ψ 2 ρ = 1. Two constraints is sufficient to solve a second or- der differential equation: having three constraints guarantees us a discrete spectrum. This is clearly a pre-requisite for a sensible model of QCD. A simple check shows that is the Green’s function to (5.5). It can also be shown that V ( q, z ′ ) of equa- tion (5.8) is given by − (1 /z ) ∂$_{z}$G ( q ; z, z ′ ) at z = ϵ . Putting this together with equation (5.8) we find 77 ψ (0) = 0 ∂$_{z}$ψ ( z$_{m}$ ) = 0 (5.13) G ( q ; $_{z, z}$′$_{) =}$ ∑ ρ ψ$_{ρ}$ ( z ) ψ$_{ρ}$ ( z ′ ) q 2 − m 2 ρ + iϵ (5.14) Π$_{V}$ ( − q $^{2}$) = − 1 g 2 5 ∑ ρ ( ψ ′ ( ϵ ) /ϵ ) 2 ( q 2 − m 2 ρ + iϵ ) m 2 ρ (5.15)