m GLYPH<Equal> 0, c GLYPH<Equal> 1.87 m GLYPH<Equal> 0.46, c GLYPH<Equal> 1.59 m GLYPH<Equal> 0.86, c GLYPH<Equal> 1.32 m GLYPH<Equal> 1.32, c GLYPH<Equal> 1.04 0 2 4 6 8 10 Ρ 0 0.5 1 1.5 Ω$_{6}$ Figure 4.2: Solutions for the ω$_{6}$ flow when ω$_{5}$ = 0 showing the dependence of the condensate on the quark mass geometry a very interesting model to study other properties of QCD. We now turn our attention to the presence of Goldstone bosons. Numerical solutions discussed in the last section have demonstrated that this system dynamically breaks the classical U (1)$_{A}$ symmetry by forming a quark con- densate. We first met Goldstone’s theorem [25] in chapter 1.2.8, and in that section we analysed the spontaneous breaking of the U (1)$_{L}$ × U (1)$_{R}$ sym- metry to U (1)$_{V}$ . We identified the η ′ (958) meson as the relevant Goldstone boson. However, matters were complicated by the presence of instantons, which made the η ′ (958) heavier than would otherwise be expected. Here, at large N, there are no instantons and so the equivalent η ′ meson should still be light. Henceforth we will describe the pseudo-Goldstone boson resulting from the spontaneous breaking of the U (1)$_{A}$ symmetry of this geometry as a pion. 4.4 Pions and their Interactions 67