F$_{ρ}$ m$_{ρ}$ 1 q $^{2}$− m 2 ρ F$_{ρ}$ m$_{ρ}$ µ ν Figure 1.5: One contribution to the vector current two-point function tons [26–28]. If we represent the vector current two-point function as figure 1.4, then one contribution to the vector two-point current will be that of figure 1.5. It can be shown [29, 30] that at large N, the mesons form an infinite, stable spectrum and that the vector two-point current is given by the sum over all meson and glueball resonances (with the correct quantum numbers), such as that in figure 1.5: The axial vector current two-point function at large N, Π$_{P}$ ( − q $^{2}$), is almost identical: (1.31) also possesses a U (1)$_{L}$ ⊗ U (1)$_{R}$ symmetry which we have yet to men- tion. Whereas the SU (2) symmetries mixed the up and down quarks, the U (1) symmetries act to add a phase to the entire spinor. Classically, we have 22 Axial Current Anomaly Π$_{V}$ ( − q $^{2}$) ≡ − ∑ ρ 〈 0 | J$_{µ}$ | ρ 〉〈 ρ | J $^{µ}$| 0 〉 ( q 2 − m 2 $_{ρ}$) = − ρ F 2 V ( q 2 − m 2 $_{ρ}$) m 2 ρ (1.38) Π$_{P}$ ( − q $^{2}$) = − ∑ a 1 F 2 P ( q 2 − m 2 a $_{1}$) m 2 a 1 (1.39)