59 Chapter 2: Motivation (a) (b) Figure 2.13: (a) Open string with i and j Chan-Paton degrees of freedom. (b) Open string scattering amplitude; the Chan-Paton index for the right end of each string becomes that of the left end of the next. In the hadronic string theory this corresponds to 3 meson to 2 meson scattering. Figures adapted from [323, 324]. be reliably approximated by the Born-Infeld action [322, 325] for the massless modes of the string theory. For Type IIB the boson part of the action on the D brane worldvolume reads: where the θ i are the angular coordinates of the 5-sphere. The r and θ i are then six scalar fields that are functions of the brane coordinates. Notice that S$_{D}$$_{3}$ = 1 (2 π ) $^{3}$g$_{s}$ α ′ 2 ∫ d $^{4}$xf − 1 [ $^{√}$− det( η$_{αβ}$ + f ∂$_{α}$r∂$_{β}$r + r $^{2}$f g$_{ij}$ ∂$_{α}$θ $^{i}$∂$_{β}$ θ j + 2 πα ′ $^{√}$f F$_{αβ}$ ) − 1$^{]}$ , (2.42) f = 4 πg$_{s}$α ′ $^{2}$N r 4 , (2.43)