content_image/308166.jpg content_image/308165.jpg content_image/308164.jpg content_image/308163.jpg content_image/308162.jpg content_image/308161.jpg
which on comparison with equation 1.38 of section 1.2.8, in the limit ϵ → 0, allows us to extract the decay constant F$_{ρ}$ : 

## 5.3.2 Axial Vector Mesons 

<img src='content_image/308166.jpg'>

Our analysis of the axial vector mesons closely follows that of the vector mesons. However things are complicated by the mixing of the pion field π and the longitudinal axial gauge field φ . We start with the action, expanded to quadratic order: 

<img src='content_image/308165.jpg'>

where we have defined v ( z ) = m$_{q}$z + σz $^{2}$, A = ( A$_{L}$ − A$_{R}$ ) / 2 and X = X$_{0}$ exp( i 2 π $^{a}$t$^{a}$). We make the gauge choice A$_{z}$ = 0 and A$_{µ}$ = $^{A$_{µ}$}$⊥ + ∂$_{µ}$φ . Then the equations of motion become 

<img src='content_image/308164.jpg'>

<img src='content_image/308163.jpg'>

<img src='content_image/308162.jpg'>

The a$_{1}$ meson is a spin-1 particle, and so is the solution to equation (5.18) with the same boundary conditions we imposed for the vector mesons, namely 

<img src='content_image/308161.jpg'>

