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how the probe brane lies within the spacetime. Using the U(1) freedom we have to re-parameterise u$_{5}$ and u$_{6}$ , let us set u$_{6}$ = 0, and look at the equation of motion for u$_{5}$ . Expanding the square root to second order, the Euler-Lagrange equation of motion is 

whose solution tends to m + cρ − 2 as ρ → ∞ . 

<img src='content_image/325478.jpg'>

From (3.6) we can see that u$_{5}$ has dimensions of mass. Hence the pa- rameter m corresponds to a mass for the ¯ qq operator, and the parameter c , with mass dimension three, corresponds to a VEV for the ¯ qq bilinear. 

Now we can investigate whether this system does or does not allow a quark condensate. For the field theory to be consistent, we require two things of the D7 flow: 

• The field theory must have a unique description at any given energy: hence the sum u 2 5 + ρ 2 must be monotonic as ρ varies from 0 to ∞ 

• The metric (3.7) is symmetric under ρ → − ρ , which implies that well- behaved solutions must obey u ′ $_{5}$(0) = 0 and u$_{5}$ (0) ̸ = ±∞ . Were it otherwise, the brane would have a kink, which would give an infinite contribution to the action 

Figure 3.2 shows three flows for m = 1 with c = − 1 , 0 and 1. As expected for a supersymmetric theory, a condensate is not allowed to form. 

If the reader is not satisfied with this numerical explanation, we can provide an analytical proof. (3.8) can be solved analytically, for all ρ . Its 

