c GLYPH<Equal> 1 c GLYPH<Equal> 0 c GLYPH<Equal> GLYPH<Minus> 1 0 2 4 6 8 10 12 14 Ρ 0.5 1.0 1.5 2.0 u$_{5}$ Figure 3.2: D7-brane flows for m = 1 with several different condensate values in AdS$_{5}$ × S 5 background. Only the c = 0 solution allows a consistent theory solution is with φ = arccos( B $^{2}$− i 2 / $^{3}$( − $_{1+}$√$_{3)}$$_{r}$ 2 B $^{2}$+ i 2 / 3 $_{(1+}$√$_{3)}$$_{r}$ 2 ). F ( φ, m ) is the incomplete elliptic integral of the first kind, using the convention defined in Mathematica version 5.2. The function is complex unless B = 0, meaning that the only well- behaved solution is when u$_{5}$ is equal to a constant. This is the solution with a zero valued condensate. By allowing m = 0 we have clearly broken the U(1) chiral symmetry that existed before. We showed in section 1.2.8 that QCD breaks chiral symmetry both by the spontaneous formation of a chiral condensate, and to a lesser extent, by the mass of the quarks. In this pure AdS system, the only way we can break chiral symmetry is by making the quarks massive. 51 u$_{5}$ ( ρ ) = A + i 2 / $^{3}$B 2 . 3 1 / 4 √$_{B}$ 6 − r 6 F ( φ, 1 4 $_{(2 +}$√$_{3))}$ × √ ( B 2 + i 2 / $^{3}$r$^{2}$)( B 4 − i 2 / $^{3}$B$^{2}$r2 + i 4 / $^{3}$r$^{4}$) (3.9)