Baryons in holographic QCD

We study baryons in holographic QCD with $\mathrm{D}4/\mathrm{D}8/\overline{\mathrm{D}8}$ multi-D-brane system. In holographic QCD, the baryon appears as a topologically nontrivial chiral soliton in a four-dimensional effective theory of mesons. We call this topological soliton brane-induced Skyrmion. Some review of $\mathrm{D}4/\mathrm{D}8/\overline{\mathrm{D}8}$ holographic QCD is presented from the viewpoints of recent hadron physics and QCD phenomenologies. A four-dimensional effective theory with pions and $\rho$ mesons is uniquely derived from the non-Abelian Dirac-Born-Infeld (DBI) action of D8 brane with D4 supergravity background at the leading order of large $N_c$, without small amplitude expansion of meson fields to discuss chiral solitons. For the hedgehog configuration of pion and $\rho$-meson fields, we derive the energy functional and the Euler-Lagrange equation of brane-induced Skyrmion from the meson effective action induced by holographic QCD. Performing the numerical calculation, we obtain the soliton solution and figure out the pion profile $F(r)$ and the $\rho$-meson profile $\tilde{G}(r)$ of the brane-induced Skyrmion with its total energy, energy density distribution, and root-mean-square radius. These results are compared with the experimental quantities of baryons and also with the profiles of standard Skyrmion without $\rho$ mesons. We analyze interaction terms of pions and $\rho$ mesons in brane-induced Skyrmion, and find a significant $\rho$-meson component appearing in the core region of a baryon.

Figure 1

Multi D-brane configurations of the $\mathrm{D}4/\mathrm{D}8/\overline{\mathrm{D}8}$ holographic QCD. $N_c$-folded D4 branes and $N_f$-folded $\mathrm{D}8\mathrm{\text{−}}\overline{\mathrm{D}8}$ branes. Flat four-dimensional space-time $x_{0–3}$ are not plotted. Gluons and quarks appear as the fluctuation modes of 4-4, 4-8 and $4\mathrm{\text{−}}\overline{8}$ strings shown by the waving lines.

Figure 2

Probe D8 branes with D4 supergravity background (4). Flat four-dimensional space-time $x_{0–3}$ are not plotted. The radial coordinate $U$ in the extra five dimensions $x_{5–9}$ is bounded from below by a horizon $U_{\mathrm{KK}}$ as $U\ge U_{\mathrm{KK}}$. Color-singlet mesons appear as the fluctuation modes of 8-8 string shown by the waving line. The two “corns” shown above are continuously connected along the angle directions ${\Omega }_4$ in $x_{5–9}$. For the argument of SO(5) singlet modes around ${\Omega }_4$, only single “corn” is to be considered, as shown in Fig. 3.

Figure 3

Coordinates $(U,\tau )$ and $(y,z)$ on the D4 supergravity solution (4); coordinates $(y,z)$ are introduced as $\frac{y}{U_{\mathrm{KK}}}\equiv (\frac{U^3}{U_{\mathrm{KK}}^3}-1{)}^{1/2}\cos (\frac{2\pi }{\delta \tau }\tau )$ and $\frac{z}{U_{\mathrm{KK}}}\equiv (\frac{U^3}{U_{\mathrm{KK}}^3}-1{)}^{1/2}\sin (\frac{2\pi }{\delta \tau }\tau )$. Stabilized D8 brane configuration (15) with the curved space-time background is also shown.

Figure 4

Chiral profile $F(r)$ and rescaled $\rho$-meson profile $\widehat{G}(r)$ of the brane-induced Skyrmion as the hedgehog soliton solution in the holographic QCD. The dashed profile in the upper figure shows the chiral profile of the standard Skyrmion without $\rho$ mesons.

Figure 5

Energy contributions of all the terms in the action (95) of brane-induced Skyrmion. Those of the standard Skyrme model with the chiral and Skyrme terms are also shown for comparison.

Figure 6

Total energy densities $4\pi r^2\epsilon (r)$ per BPS value of $12{\pi }^2$ for the brane-induced Skyrmion (solid curve) and the standard Skyrmion (dashed curve).

Figure 7

Contributions of all the terms in the effective action (95) to the energy density $4\pi r^2\epsilon (r)/12{\pi }^2$ of the brane-induced Skyrmion.

Figure 8

Contributions of $\rho$-meson interaction terms in the effective action (95) to the energy density $4\pi r^2\epsilon (r)/12{\pi }^2$ of the brane-induced Skyrmion with the total energy density.