content_image/305062.jpg content_image/305061.jpg content_image/305063.jpg content_image/305064.jpg
Applying the Euler-Lagrange equation of motion for Φ we find 

<img src='content_image/305062.jpg'>

There are two important points to be noted in equation (4.22). 

<img src='content_image/305061.jpg'>

<img src='content_image/305063.jpg'>

Firstly there is a an explicit Φ → e $^{iα}$Φ symmetry. In the massless case this is the U (1)$_{A}$ symmetry on the quarks. 

Secondly, in the UV limit ( r → ∞ ), the equation of motion simplifies to 

<img src='content_image/305064.jpg'>

which has the simple solution Φ = m + cρ − $^{2}$. The two integration constants, m and c , correspond to a quark mass and a VEV for the quark bilinear ¯ qq respectively. 

Returning to the full equation of motion, we look for regular solutions that have the UV boundary condition σ = m + cρ − $^{2}$. (For the moment θ is turned off: it corresponds to the pion fields, which we will demonstrate later). A few of the solutions are plotted in figure 4.2. 

The key feature of figure 4.2 is the m = 0 , c = 1 . 86 curve. This shows that this geometry demonstrates chiral symmetry breaking: there is a quark condensate even if the quarks are massless! This makes the Constable-Myers 

