Strong force fields and stabilities of the nucleon and singly heavy baryon ${\mathrm{\Sigma }}_c$

We investigate the strong force fields and stabilities of the nucleon and the singly heavy baryon ${\mathrm{\Sigma }}_c$ within the framework of the chiral quark-soliton model. Having constructed the pion mean fields in the presence of the $N_c-1$ level quarks self-consistently, we are able to examine the gravitational form factors of ${\mathrm{\Sigma }}_c$. We mainly focus in the present work on the stability conditions for both the nucleon and ${\mathrm{\Sigma }}_c$ and discuss the strong force fields and their physical implications. We also present the results for the gravitational form factors and relevant observables, emphasizing the difference between the nucleon and ${\mathrm{\Sigma }}_c$.

Figure 1

The comparison of the $N_c-1$ light-quark energy density for the singly heavy baryon ${\mathrm{\Sigma }}_c$ with the full energy density for the nucleon. The left panel depicts the energy densities as functions of radius $r$, whereas the right panel shows the energy densities multiplied by $4\pi r^2$ as functions of $r$. The solid and dashed curves draw, respectively, $ϵ(r)$ for ${\mathrm{\Sigma }}_c$ and for the nucleon.

Figure 2

The comparison of the $N_c-1$ light-quark density for the total angular momentum of the singly heavy baryon ${\mathrm{\Sigma }}_c$ with that for the nucleon. The left panel depicts the total angular-momentum densities as functions of radius $r$, whereas the right panel shows the total angular-momentum densities multiplied by $4\pi r^2$ as functions of $r$. The solid and dashed curves draw, respectively, ${\rho }_J(r)$ for ${\mathrm{\Sigma }}_c$ and for the nucleon.

Figure 3

The comparison of the pressure distributions for ${\mathrm{\Sigma }}_c$ with those for the nucleon. The numerical results for the pressure densities of the nucleon and ${\mathrm{\Sigma }}_c$ are drawn as functions of radius $r$. The solid and dashed curves depict, respectively, those for ${\mathrm{\Sigma }}_c$ and for the nucleon.

Figure 4

The left panel presents decomposition of the level and Dirac-continuum contributions to the pressure. The right panel shows $r^2$-weighted pressures for both the light and singly heavy baryons.

Figure 5

Comparison of the results for the shear-force density $s(r)$ and $p_r(r)$ of ${\mathrm{\Sigma }}_c$ with those for the nucleon. The solid curves depict those of ${\mathrm{\Sigma }}_c$, whereas the dashed ones draw those of the nucleon.

Figure 6

Results for the normal and tangential force fields in the nucleon and ${\mathrm{\Sigma }}_c$. In the upper left panel, the results for the level and Dirac-continuum contributions to the normal force field in the nucleon are drawn in the short-dashed and long-dashed curves, respectively. In the upper right panel, those in ${\mathrm{\Sigma }}_c$ are drawn in the same notation. In the lower left (right) panel, we depict the results for the tangential force field in the nucleon (${\mathrm{\Sigma }}_c$) in the same notation.

Figure 7

Comparison of the normal and tangential force fields in the nucleon with those of ${\mathrm{\Sigma }}_c$. The solid curves depict those of ${\mathrm{\Sigma }}_c$, whereas the dashed ones draw those of the nucleon.

Figure 8

Visualization of the normal force fields $4\pi r^2{T}^{ij}{\mathit{e}}_r^j$ and tangential force fields $4\pi r^2{T}^{ij}{\mathit{e}}_{\varphi }^j$. In the upper panel, we visualize the normal force fields in the nucleon (left) and ${\mathrm{\Sigma }}_c$ (right), respectively. In the lower panel, we show the tangential force fields in the nucleon (left) and ${\mathrm{\Sigma }}_c$ (right), respectively. The radius of the disc is taken to be 1.5 fm; the color legend gives the absolute value of the force fields in $\mathrm{GeV}/\mathrm{fm}$.

Figure 9

In the upper panel, the infinitesimal force fields $d{F}_{(r,\theta ,\varphi )}$ defined in Eq. (21) are visualized as the arrows, which will be used in the 3D visualization of the strong force fields in Figs. 10, 11, and the lower panel of Fig. 9. In the lower left and right panels, the 3D visualization of the strong force fields ($\mathit{F}$) for the nucleon and ${\mathrm{\Sigma }}_c$ are, respectively, illustrated.

Figure 10

3D visualization of the strong force field ($\mathit{F}$) as a vector field inside a nucleon. We display the 3D force field exerting locally at each point in given shells with the distance from the center varied from 0.2, 0.45, 0.6, and 1 fm. Note that at 0.45 fm the tangential force field vanishes.

Figure 11

3D visualization of the strong force field ($\mathit{F}$) as a vector field inside ${\mathrm{\Sigma }}_c$. We display the 3D force field exerting locally on each point in given shells with the distance from the center varied from 0.2, 0.35, 0.6, and 1 fm. Note that at 0.35 fm the tangential force field vanishes.

Figure 12

The results for $p(ϵ)$, $s(ϵ)$, $p_r(ϵ)$, and $p_{\varphi }(ϵ)$. The solid curves depict those of ${\mathrm{\Sigma }}_c$, whereas the dashed ones draw those of the nucleon.

Figure 13

Results for the gravitational form factors $A(t)$, $D(t)$, and $J(t)$. The solid curves depict those of ${\mathrm{\Sigma }}_c$, whereas the dashed ones draw those of the nucleon.