Figure 4.3: Quark mass versus meson mass, showing the massless pion when m$_{q}$ = 0. We plot the square root of the quark mass to demonstrate the linear relationship. 0.02 0.04 0.06 0.08 0.10 GLYPH<Radical1> GLYPH<Radical1Extens> GLYPH<Radical1Extens> GLYPH<Radical1Extens> GLYPH<Radical1Extens> m 0.05 0.10 0.15 0.20 0.25 MR 4.4.2 The Vector Meson Equation Of Motion There is an additional field present on the D7 brane which we have left undiscussed: it is the gauge field partner of the scalar Φ. The field strength of this gauge field enters in the standard way as 2 πα ′ F ab in the square root. The leading Lagrangian for this field, in a background configuration σ$_{0}$ is There is an additional term that could be added to the DBI action: the Wess-Zumino term, which gives the coupling of the four-form C (4) to the gauge fields. It is not included here because when we calculate the equations of motion for the gauge fields, this term is only relevant for the gauge fields with a vector index on the S $^{3}$. We are only interested in states that carry L = 2 π $^{2}$R4 (2 π ) $^{7}$α ′ $^{4}$g$_{s}$ ∫ dρ e φ G (√ 1 + ( ∂$_{ρ}$σ$_{0}$ ) $^{2}$H ( ( ρ 2 + σ 2 $_{0}$) 2 − 1 ( ρ 2 + σ 2 $_{0}$) 2 + 1 ) 1 4 (2 $_{πα}$′$_{)}$ $_{2}$1 4 F $^{µν}$F$_{µν}$ + 1 2 R 2 1 √ 1 + ( ∂$_{ρ}$σ$_{0}$ ) 2 ( ( ρ 2 + σ 2 $_{0}$) 2 − 1 ( ρ 2 + σ 2 $_{0}$) + 1 ) 1 2 ( ρ 2 + σ 2 $_{0}$) 2 ( ρ 2 + σ 2 $_{0}$) 2 − 1 F $^{µρ}$F$_{µρ}$ (4.27) 69