Generalized Skyrme model with the loosely bound potential

We study a generalization of the loosely bound Skyrme model which consists of the Skyrme model with a sixth-order derivative term—motivated by its fluidlike properties—and the second-order loosely bound potential—motivated by lowering the classical binding energies of higher-charged Skyrmions. We use the rational map approximation for the Skyrmion of topological charge $B=4$, calculate the binding energy of the latter, and estimate the systematic error in using this approximation. In the parameter space that we can explore within the rational map approximation, we find classical binding energies as low as 1.8%, and once taking into account the contribution from spin-isospin quantization, we obtain binding energies as low as 5.3%. We also calculate the contribution from the sixth-order derivative term to the electric charge density and axial coupling.

Figure 1

Parameter space in the loosely bound Skyrme model with the BPS-Skyrme term. $c_6$ is the BPS-Skyrme term coefficient and $m_2$ is the coefficient of the loosely bound potential. The region $A$ corresponds to a small BPS-Skyrme term and the region of parameter space that we will study in this paper. $B$ corresponds to a large $m_2$ and medium-sized value of $c_6$. Finally, $C$ corresponds to the near-BPS regime of the BPS-Skyrme model; in this regime, the kinetic term and the Skyrme term are small perturbations of the BPS-Skyrme model.

Figure 2

Relative classical binding energy, ${\delta }_4$, in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 3

Relative total binding energy, ${\delta }_4^{\mathrm{tot}}$, in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 4

Relative (a) classical and (b) total binding energy, ${\delta }_4$ and ${\delta }_4^{\mathrm{tot}}$, as functions of $m_2$ for $m_1=0.25$. The blue circles are full PDE calculations from Ref. [31], the red dashed line is the calculation using the rational map approximation, and finally, the green solid line is the rational map approximation with a correction for systematic error, see Eq. (47).

Figure 5

Relative total binding energy with a linear correction of the systematic error due to the rational map approximation, ${\delta }_4^{\text{corrected}}$, in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 6

Pion decay constant, $f_{\pi }$ [MeV], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 7

Skyrme term coefficient, $e$, in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 8

Nucleon mass, ${\tilde{m}}_N$ [MeV], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 9

Mass of the $\mathrm{\Delta }$ resonance, ${\tilde{m}}_{\mathrm{\Delta }}$ [MeV], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 10

Pion mass, ${\tilde{m}}_{\pi }$ [MeV], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$. The two bold red lines indicate the physical values of the pion masses.

Figure 11

Baryon charge radius of the proton, $r_1$ [fm], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 12

Electric charge radius of the proton, $r_{E,1}$ [fm], in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.

Figure 13

Baryon and electric charge densities, $\frac{1}{2{\pi }^2}{\mathcal{B}}^0$ as red dashed lines and ${\rho }_E$ as blue solid lines, in various points of the parameter space: $(m_2,c_6)=(0.5,0),(0.5,0.8),(0,0),(0,0.8)$.

Figure 14

Axial coupling, $g_A$, in (a) the $(m_2,c_6)$ plane for $m_1=0.25$ and (b) the $(m_1,c_6)$ plane for $m_2=0.7$.