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## 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. 

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