The generalisation of equation (5.8) is Using the fact v ( q, r ′ ) = [ K$_{1}$ ( r ) ∂$_{r}$G ( q ; r, r ′ $^{)]$_{r}$$_{=}$}$∞, we find Comparing this to (6.11) we identify the rho decay constant as: The axial vector field is defined as A a µ = ( A a Lµ − A a $_{Rµ}$). We choose the gauge such that A$_{ar}$ ( x, r ) = 0 and A µ a = A µ a ⊥ + ∂ $^{µ}$φ . There are three equations of motion: one for each of the a$_{1}$ meson field, the π field and the longitudinal axial gauge field φ : where K$_{3}$ ( r ) = Hf 3 / 2 − δ/ 2+Δ / $^{2}$h$^{3}$r$^{3}$(1 + ˙ σ $^{2}$) − 1 / 2 and K$_{4}$ ( r ) = f 1+Δ / $^{2}$h$^{2}$r$^{3}$(1 + ˙ σ $^{2}$) − 1 / $^{2}$, and we have separated $^{A}$⊥ as $^{A$_{i}$}$⊥ ( x, r ) = A ( q, r ) exp( iqx ). Looking for the normalizable solutions to (6.16) requires us to set the boundary conditions ψ$_{a}$$_{1}$ ( ∞ ) = 0 and ∂$_{r}$ψ$_{a}$$_{1}$ (0) = 0. We can then predict the a$_{1}$ meson mass spectrum. And in an almost identical analysis to that 92 6.2.2 Axial sector Π$_{V}$ ( − q $^{2}$) = [ 1 g 2 $_{5}$q $_{2}$K$_{1}$ ( r ) ∂$_{r}$v ( q, r ) ] r = ∞ (6.13) Π$_{V}$ ( − q $^{2}$) = − lim r →∞ 1 g 2 5 ∑ ρ ( K$_{1}$ ( r ) ψ ′ $_{ρ}$( r )) 2 ( q 2 − m 2 $_{ρ}$) m 2 ρ (6.14) F 2 ρ = lim r →∞ 1 g 2 5 ( K$_{1}$ ( r ) $_{ψ}$′ $_{ρ}$( r ) ) 2 (6.15) [ ∂$_{r}$ ( K$_{1}$ ( r ) ∂$_{r}$A a $_{µ}$( r )) + q $^{2}$K$_{2}$ ( r ) A a $_{µ}$( r ) − g 2 $_{5}$σ ( r ) $^{2}$K$_{3}$ ( r ) A a $_{µ}$( r ) ] ⊥ = 0(6.16) a ∂$_{r}$ ( K$_{1}$ ( r ) ∂$_{r}$φ ) + g$_{5}$ $^{2}$σ ( r ) $^{2}$K$_{3}$ ( r )( π − φ $^{a}$) = 0(6.17) − q $^{2}$K$_{1}$ ( r ) ∂$_{r}$φ + g$_{5}$ $^{2}$K$_{4}$ ( r ) σ ( r ) $^{2}$∂$_{r}$ π = 0(6.18)