4.4.1 The Pion Equation Of Motion In the case of one flavour, the pion is simply a bound ¯ qq state, and will be described by the field Φ. The Goldstone fluctuation corresponds to os- cillations of the field along the vacuum manifold, which in the chiral limit means fluctuations in the angular θ direction of Φ. To leading order, if we have a background configuration for the D7 brane described by σ$_{0}$ , we can look at small fluctuations in the θ direction. The action for these fluctuations (to quadratic order) is given by ex- panding the DBI action (4.19): And the resulting Euler-Lagrange equation of motion is: We look for values of MR as a function of m that give regular solutions to (4.26). The UV boundary condition we impose is f = 1 /ρ $^{2}$, which reflects the fact that pion has the same UV scaling dimension as ¯ qq . The results are plotted in figure 4.3. There is a massless pion when m$_{q}$ = 0, in accordance with Goldstone’s theorem. 68 M $^{2}$R$^{2}$e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) $_{2}$H ( ( ρ 2 + σ 2 $_{0}$) 2 + 1 ( ρ 2 + σ 2 $_{0}$) 2 − 1 ) (1 − δ ) / 2 ( ρ 2 + σ 2 $_{0}$) 2 − 1 ( ρ 2 + σ 2 $_{0}$) 2 σ 2 $_{0}$f + d dρ ( e $^{φ}$G √ 1 + ( ∂$_{ρ}$σ 2 $_{0}$) $_{2}$σ 2 $_{0}$∂$_{ρ}$ f ) = 0 (4.26) S = 2 π 2 (2 π ) $^{7}$α ′ $^{4}$g$_{s}$ ∫ dρ d $^{4}$x e $^{φ}$R $^{4}$G √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) 2 × ( 1 + 1 2 g $^{ρρ}$g$_{θθ}$ ( ∂$_{ρ}$θ ) 2 1 + ( ∂$_{ρ}$σ$_{0}$ ) 2 + 1 2 g $^{µν}$g$_{θθ}$ ∂$_{ν}$θ∂$_{µ}$θ 1 + ( ∂$_{ρ}$σ$_{0}$ ) 2 ) (4.25)