6.2 Vector and Axial Mesons 6.2.1 Vector sector In exactly the same way that we calculated the vector and axial meson masses and decay constants in chapter 5, we repeat here. Just as in section 5.3.1, we look for solutions to the vector equation of motion that are of the form V a $_{µ}$( x, r ) = V a $_{µ}$( r ) exp( iqx ) with the gauge V a r ( x, r ) = 0. We find with K$_{1}$ = f 1 / $^{2}$hr$^{3}$(1 + ˙ σ $^{2}$) − 1 / 2 and K$_{2}$ = Hf 1 − δ/ $^{2}$h$^{2}$r$^{3}$(1 + ˙ σ $^{2}$) − 1 / 2 The ρ mesons are interpreted as the normalizable modes of (6.10), mean- ing that we impose the boundary conditions V ′ (0) = 0 to ensure the smooth- ness of the solution, and V ( ∞ ) = 0 to ensure the normalization is finite. Our analysis for the decay constants follows the same path as that laid out in section 5.3.1. For large N, the vector current correlator can be written as the sum over ρ resonances: In order to find F$_{ρ}$ , we proceed by finding the Green’s function solution to (6.10). Imposing the completeness relation ∑ $_{ρ}$K$_{2}$ ( r ) ψ$_{ρ}$ ( r ) ψ$_{ρ}$ ( r ′ ) = δ ( r − r ′ ) on the set of eigenfunctions one finds 91 ∂$_{r}$ ( K$_{1}$ ( r ) ∂$_{r}$V a $_{µ}$( r ) ) + q $^{2}$K$_{2}$ ( r ) V a $_{µ}$( r ) = 0 (6.10) Π$_{V}$ ( − q $^{2}$) = − ∑ ρ F 2 ρ ( q 2 − m 2 $_{ρ}$) m 2 ρ (6.11) G ( q ; $_{r, r}$′$_{) =}$ ∑ ρ ψ$_{ρ}$ ( r ) ψ$_{ρ}$ ( r ′ ) q 2 − m 2 ρ (6.12)