Modifying the pion mass in the loosely bound Skyrme model

We study the loosely bound Skyrme model with the addition of two different pion mass terms; this is the most general potential of polynomial form up to second order in the trace of the Skyrme field. The two pion mass terms are called the standard pion mass term and the modified pion mass term. We find that the binding energies are not reduced by the introduction of the modified pion mass, but it is analogous to the standard pion mass term with a decrease in the value of the mass parameter of the loosely bound potential (for large values of the latter parameter). We find by increasing the overall pion mass that we can reduce the classical binding energy of the 4-Skyrmion to the 2.7% level and the total binding energy including the contribution from spin-isospin quantization is reduced to the 5.8% level.

Figure 1

Classical binding energy ${\delta }_4$ as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 2

Calibration of the pion decay constant, $f_{\pi }$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 3

Calibration of the Skyrme term coefficient, $e$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 4

Pion mass in physical unit, ${\tilde{m}}_{\pi }$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 5

Nucleon mass in physical unit, ${\tilde{m}}_N$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 6

Mass of the $\mathrm{\Delta }$ resonance in physical unit, ${\tilde{m}}_{\mathrm{\Delta }}$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 7

Charge radius of the proton in physical unit, ${\tilde{r}}_1$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 8

Total binding energy, ${\delta }_4^{\text{tot}}$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5.

Figure 9

Breakdown of the total binding energy, ${\delta }_4^{\text{tot}}$, as function of ${\mathbf{m}}_2^2$; four series of points are shown corresponding to $m_{\pi }=0.125$, 0.25, 0.375, 0.5, corresponding to (a),(b),(c),(d), respectively. The bottom of the arrows corresponds to the classical contribution, whereas the arrow head includes the quantum correction.

Figure 10

The columns show $m_{\pi }=0.125$ Skyrmion solutions with $m_2=0.7$, 0.8, 0.9 and the rows correspond to $\alpha$ from 0 to 1 in steps of 0.2 from top to bottom.

Figure 11

The columns show $m_{\pi }=0.25$ Skyrmion solutions with $m_2=0.7$, 0.8, 0.9 and the rows correspond to $\alpha$ from 0 to 1 in steps of 0.2 from top to bottom.

Figure 12

The columns show $m_{\pi }=0.375$ Skyrmion solutions with $m_2=0.9$, 1, 1.1 and the rows correspond to $\alpha$ from 0 to 1 in steps of 0.2 from top to bottom.

Figure 13

The columns show $m_{\pi }=0.5$ Skyrmion solutions with $m_2=1.1$, 1.2, 1.3 and the rows correspond to $\alpha$ from 0 to 1 in steps of 0.2 from top to bottom.