Skyrmions with holography and hidden local symmetry

We study baryons as Skyrmions in holographic QCD with $\mathrm{D}4/\mathrm{D}8/\overline{\mathrm{D}8}$ multi-D brane system in type IIA superstring theory, and also in the nonlinear sigma model with hidden local symmetry. Comparing these two models, we find that the extra dimension and its nontrivial curvature can largely change the role of (axial) vector mesons for baryons in four-dimensional space-time. In the hidden local symmetry approach, the $\rho$-meson field as a massive Yang-Mills field has a singular configuration in Skyrmion, which gives a strong repulsion for the baryon as a stabilizer. When the $a_1$ meson is added in this approach, the stability of Skyrmion is lost by the cancellation of $\rho$ and $a_1$ contributions. On the contrary, in holographic QCD, the $\rho$-meson field does not appear as a massive Yang-Mills field due to the extra dimension and its nontrivial curvature. We show that the $\rho$-meson field has a regular configuration in Skyrmion, which gives a weak attraction for the baryon in holographic QCD. We argue that Skyrmion with $\pi$, $\rho$, and $a_1$ mesons become stable due to the curved extra dimension and also the presence of the Skyrme term in holographic QCD. From this result, we also discuss the features of our truncated-resonance analysis on baryon properties with $\pi$ and $\rho$ mesons below the cutoff scale $M_{\mathrm{KK}}\sim 1\mathrm{GeV}$ in holographic QCD, which is compared with other 5D instanton analysis.

Figure 1

$F(r)$ and $G(r)$ of Skyrmion in the HLS approach with various values of $a$ parameters in (8).

Figure 2

Schematic figure of Skyrmion in HLS. Dashed black and grey lines show the configurations of $F(r)$ and $G(r)$ without interactions, respectively. $G(r)$ appears as broad singular configuration with the boundary $G(0)=2$.

Figure 3

Schematic figure of holographic duality between gauge theory as QCD on D brane and supergravity around D brane. Masses of mesons $m_{n=1,2,3,\cdots }$ are given by the oscillation of meson wave functions ${\psi }_n(z)$ in the extra dimension $z$ [11]. Therefore, the curvature in the extra dimension around the D brane is essential to give the discrete meson mass spectra as QCD on the D brane. Then, regions A and B have different curvatures, and the gauge symmetry on A will be distorted on B except for the gauge symmetry of QCD, giving the relation $g_{3\rho }^2/g_{4\rho }\ne 1$.

Figure 4

$F(r)$ and $G(r)$ of Skyrmion in holographic QCD. Dashed lines with label “$S_{\pi \mathrm{\text{−}}\rho }=0$” show the configurations without interactions between $\pi$ and $\rho$ in (19).

Figure 5

Schematic figure of Skyrmion in holographic QCD. Dashed black and grey lines show the configurations of $F(r)$ and $G(r)$ without interactions, respectively. $G(r)$ appears as regular configuration in the core region of a baryon with the boundary $G(0)=0$.