From the Sakai-Sugimoto model to the generalized Skyrme model

We derive the generalized Skyrme model as a low-energy effective model of the Sakai-Sugimoto model. The novelty with the past is the presence of the sextic term equal to the topological charge squared. This term appears when the $\omega$ meson, and the tower of states on top of it, are integrated out. We claim that, in the small ’t Hooft coupling limit, the instanton is well described by a Skyrmion arising from the low energy effective Lagrangian of the Sakai-Sugimoto model. The sextic term plays a dominant role in this limit. Moreover, when a pion mass term is added, we recover the BPS Skyrme model in the small ’t Hooft coupling limit.

Figure 1

Skyrmion profile for decreasing values of $\mathrm{\Lambda }$. As can be seen, the size of the soliton solution increases as $\mathrm{\Lambda }$ becomes small.

Figure 2

Skyrmion’s profile functions rescaled to the same size. In red, we have plotted the rescaled solution of (3.8), in shades of blue, the ones of the full massless model.

Figure 3

Energy for a range of values of $\mathrm{\Lambda }$ in the massless model. Black dots are the energy obtained with the model developed. The dashed red and green lines correspond, respectively, to fitting linear and square root relations, as are expected to develop in the large and small $\mathrm{\Lambda }$ regions. The dashed blue line represents the energy of the BPST instanton, which becomes the correct description of the baryon at large $\mathrm{\Lambda }$ [4]. The black star corresponds to the energy computed at the phenomenological value of $\mathrm{\Lambda }=1.568$: as can be seen, this value lies just in between the two regimes of large and small ’t Hooft coupling.

Figure 4

Skyrmion’s profile functions in the generalized model with a pion mass potential.

Figure 5

Energy for a range of values of $\mathrm{\Lambda }$. The dashed red line corresponds to a fitting linear relation, as it is expected to develop in the large $\mathrm{\Lambda }$ region. The vertical red line corresponds to the value satisfying $\mathrm{\Lambda }=R^{-3}=m_{\pi }^3$, which is expected to be the scale at which the change in behavior happens. The blue dashed line again represents the energy of the BPST configuration, which describes the baryon at large $\mathrm{\Lambda }$, with the addition of the shift in mass computed in [15] due to the quark mass term.

Figure 6

Skyrmion’s profile functions rescaled to the same size. In red, we have plotted the rescaled analytical solution of the BPS model ${\mathcal{L}}_{06}$, in shades of blue, the ones of the full massive equation (3.9).